A hole is a short tube

The vent a cone cannot place closed with a sentence that names an assumption four rungs of this ladder have been standing on:

Everything above treats an open hole as a point at which the pressure is released, and a real hole is a short tube of finite radius and finite height whose own inertance sets how completely it releases.

Under the point model, two holes at the same station are the same hole. A tone hole is where the tube ends, the note is the note of the shortened tube, and a hole’s diameter is a detail about how loud it is.

The air in a hole has mass. Pushing it out of the way costs an inertance that goes as the chimney’s height over its area, which is small for a wide short hole and large for a narrow deep one — and a hole with an inertance does not end the bore. It loads it. The tube goes on behaving as though it continued for some distance past the hole, and that distance is what a maker is choosing when they choose a drill.

The size of the thing that was being ignored #

A hole as wide as the bore itself leaves 12.5 millimetres of tube behind it. That is the hole’s own end correction and it is small, which is why the point model has survived four rungs.

A hole of 1.5 millimetres radius leaves 87 millimetres. The tube is 600 millimetres long and the hole is at 400, so a seventh of the whole instrument is sitting behind a hole that is supposedly the end of it.

The note that comes out runs from 202 hertz for the widest hole to 171 for the narrowest — 289 cents, nearly a minor third, from one station and one tube. What note a tone hole gives is not a fact about where it is.

That is not a correction to the four rungs above; it is the parameter they held fixed. Every hole in every figure of this anchor so far has been the same hole.

The same object the ladder already had, at the other end #

There is nothing new in the physics here, and saying where it came from is the fastest way to see what the point model was throwing away.

The tube ends after it ends is this collection’s essay on the open end of a pipe: the air just outside a tube’s mouth has to be pushed too, so the tube behaves as though it were about 0.6 of a radius longer than it is, and that correction is what makes a computed pipe length disagree with a measured one. An open tone hole is the same object. It is a short pipe opening sideways out of a long one, and the length it adds is its own chimney plus its own end correction, divided by how much narrower it is than the bore.

The division is the part the point model lost. A hole’s inertance goes as its height over its area, so halving a hole’s radius quadruples what it costs to push the air out of it. That fourth power is why the curve above is a curve rather than a small offset, and it is why an instrument’s smallest holes are the ones whose diameter matters most.

Seven holes that play one note #

Run it the other way. Fix the note and ask where each hole has to go.

A hole of 7.5 millimetres radius sounds 196 hertz at 411 millimetres from the throat. A hole of 1.4 millimetres sounds the same 196 hertz at 307 millimetres. The stations differ by 104 millimetres — a fifth of the instrument’s length — and in the first register the two are indistinguishable.

That is the trade a woodwind maker has. Holes have to be reachable by fingers, which sets where they can be; holes have to give the right notes, which sets where they must be; and the diameter is the free variable that reconciles the two. A hole is moved up the tube and made smaller, or moved down and made larger, and the note stays.

This is why a fingering chart looks the way it does. The holes on a recorder or a baroque flute are neither evenly spaced nor evenly sized, and their sizes vary by a factor of two down one instrument. Under the point model that is inexplicable — the stations would be set by the notes alone. Under this one it is the only way an instrument playable by a hand can also be in tune. On a cone the same trade exists and is tighter, because a cone’s modes are a full harmonic series and a hole that is wrong for the fundamental is wrong for the octave in the same direction.

And the second register tells them apart at once #

The vertical axis of that figure is what the trade actually costs, and the cost is not small.

A cylindrical tube’s second resonance sits at three times the first, which is the twelfth a clarinet overblows. Measured on the seven equivalent holes, that twelfth is 2 cents flat for the widest hole, 35 for a five-millimetre one, 139 for a three-millimetre one, and 453 cents flat — nearly a fourth — for the narrowest.

The mechanism is one line. A hole’s inertance is ω·ρ·t/(πb²), so it grows with frequency: the same hole is a bigger obstacle to the twelfth than to the fundamental, lengthens the tube more at the higher frequency, and pulls the register flat. A wide hole has almost no inertance to grow and stays out of the way at both.

So the choice a maker makes between “small hole, low down” and “large hole, high up” is a choice about the second register, and it is a large one. The first register cannot see it and the second register is nothing but it.

This is the arithmetic behind the compromise the third rung computed and the one the fourth found on a cone. Those rungs asked where one hole should go when it has to serve two jobs. This one says the diameter is a second control on exactly the same conflict, and it is the one that acts on the registers separately.

Cross-fingering, and the explanation that is wrong #

The other thing a fingering chart is full of is patterns with gaps in them: a hole open, and a hole below it closed. On a baroque woodwind that is how most of the chromatic notes are made, and it flattens the note by something between a quarter-tone and a semitone.

The standard explanation is the closed hole’s own volume. A closed hole is a stub of trapped air hanging off the tube; a stub is a compliance; adding compliance is adding volume; adding volume is lengthening the tube; the note goes flat.

That explanation predicts a definite number, and the number is wrong by a factor of a hundred.

Put one open hole in a tube and one closed hole below it, with nothing else, and the note moves by 0.26 of a cent. The stub is 4.5 millimetres of radius and 6 of height; its volume converted to an equivalent length of bore is under two millimetres; and it sits beyond the open hole where the pressure is small, so most of even that is lost. The compliance argument is quantitatively correct and it accounts for a quarter of one per cent of the effect.

Now do it on an instrument. A real fingering has every hole below the first open one open — that is what makes the first open hole first. Closing one of them is not adding a stub; it is removing a leak.

On a lattice of eight holes at 28-millimetre spacing, closing the hole immediately below the first open one takes the note 32 cents flat. Closing the first two takes it 55. Closing three takes it 72. Those are the sizes of the accidentals on a baroque recorder, and they come out of a model with no fitted parameters in it.

A cross-fingering is the tube below the first open hole ceasing to leak, not a bottle being added to it. The two explanations differ by two orders of magnitude and only one of them is a mechanism.

Which is why a keyed instrument cannot do it #

The consequence is a historical one and the model is unambiguous about it.

If the flattening comes from the lattice ceasing to leak, it must depend on how much the lattice was leaking, which is hole size. Sweeping it: the cross-fingering is worth 55 cents on holes of 1.8 millimetres radius, 32 on 3.5, and 16 on 6.5.

Small holes are what a fingertip can cover directly. Large holes need a pad, and a pad needs a key. So the instrument that cross-fingers well is precisely the instrument that has no keys, and the reason is not that the keys got in the way — it is that the holes had to be enlarged before the keys could exist, and enlarging the holes is what destroyed the cross-fingerings.

That is the whole nineteenth-century woodwind argument in one curve. Boehm’s flute enlarged and regularised the holes to make every note equally strong and equally in tune, and it needed a key for every one of them because they were too big and too far apart for fingers. What it lost — the shaded, differently-coloured cross-fingered notes that eighteenth-century players used expressively — is the same quantity, spent.

The usual telling has the loss as a side effect of the mechanism. It is a side effect of the hole diameter, and the mechanism is a side effect of that.

Which computation produced the numbers #

The tube is a 60-centimetre cylinder of 7.5-millimetre radius — a clarinet’s bore, near enough — solved as a transmission line from the open far end back to the throat in 300 sections, with Benade’s visco-thermal wall losses and a baffled-piston radiation load at the mouth.

Each hole is a shunt: at its station, the impedance looking down the bore is put in parallel with the hole’s own. An open hole’s impedance is the inertance of the air in its chimney, taken as the hole’s height plus 1.5 radii, in series with the radiation load of a piston of the hole’s own size. A closed hole’s is a stub of trapped air, which is a compliance. The radiation term is what allows a lattice to have a cutoff rather than merely a corner, and it is the same convention the first rung of this ladder uses.

A fingering’s note is the first impedance maximum, which is what a reed drives. The peak is interpolated parabolically in log frequency rather than read off the sweep, and that is not a nicety: at 620 logarithmic points over four octaves the grid is eight cents wide at the bottom, and the first version of the cross-fingering control came back at exactly zero seven times in a row, which looks like a finding and is a quantiser.

The substitution length is found by bisecting on the length of a plain tube until its first resonance matches the holed tube’s. The equivalent stations are found the same way, bisecting on the hole’s position until its note matches a target.

Where the model stops #

The bore is a cylinder and most woodwinds are not. An oboe, a bassoon and a saxophone are cones, and a cone’s mode spacing and hole behaviour differ — which is the fourth rung’s whole subject. The lattice arithmetic here carries over; the register numbers do not, because a cone’s second mode is the octave rather than the twelfth.

The chimney is a cylinder too. A real tone hole is undercut — reamed out inside so the opening into the bore is larger than the hole at the surface — and undercutting is precisely the maker’s way of changing a hole’s inertance without changing what the finger meets. Every result here is about a hole a maker could not adjust after drilling, and undercutting is how they adjust it.

The bore is at one temperature. Every frequency here scales with the speed of sound, so a cold instrument is a flat one — a wind instrument is a thermometer is the rung about the size of that. The ratios in this essay, which is all of its argument, are untouched by it.

There is no player. The note computed is the bore’s resonance and the note sounded is where a reed or an air jet decides to lock, which is near the resonance and not on it. A player also covers holes partially, and half-holing is a continuous control this model would take and has not been asked for.

And the lattice is regular. Eight equal holes at equal spacing is a caricature: real instruments have irregular spacing set by hands and irregular sizes set by tuning, which is the very thing the first half of this essay is about. The cross-fingering numbers should be read as the size of an effect rather than as an instrument’s chart.

What the picture cannot show #

It cannot show the tone. A cross-fingered note is famously different in colour, not merely flatter, and every eighteenth-century treatise says so. That is a claim about the spectrum, which needs the higher resonances and the radiation from each open hole weighted properly, and this figure reports one frequency.

Nor can it show the fingers. The reason holes are where they are is a hand, and no model here contains one. The trade between station and diameter is only a trade because a hand exists.

It cannot show what happens above the cutoff. Everything here is a first or second resonance, well below the lattice’s corner. Above it the holes stop being individual objects and the tube behaves as though it ended at the first one, which is the first rung’s finding and is the regime the highest notes of every woodwind live in.

It cannot show the chart as a document. A fingering chart is a notation, and what a tablature keeps is the essay about what a notation of hand positions records and what it discards. A chart records which holes are covered and never records their diameters, which this essay says is half of what decides the note.

And it cannot show the register key. The vent that makes a woodwind overblow is a tone hole too, and everything here applies to it — a narrow register hole should pull the second register flat by exactly the amount computed above, which is the kind of number the third rung’s compromise was trading against and did not have.

Whose instruments, and when #

The dimensions are generic rather than any instrument’s: a clarinet-sized cylinder, holes from 1.5 to 7.5 millimetres in radius, chimneys of three to six millimetres, spacings of 28.

The historical claim is the one about hole size and it is narrow. Renaissance and baroque woodwinds have small holes covered directly by fingers, and their chromatic notes are cross-fingerings. Nineteenth-century keyed woodwinds have large holes covered by pads, and their chromatic notes are holes of their own. The model says the first is a consequence of the second and prices it at a factor of three and a half.

What it does not say is that anybody made the trade deliberately. Boehm was explicit that he was after evenness and a strong sound, and the cross-fingerings were not something he priced and gave up; they were something that stopped working. The players who complained about the new flute complained about exactly this, and were told they were nostalgic.

Where this ladder goes next #

Five rungs. Above a certain note the holes stop working; one hole does a dozen jobs; the register hole spoils a note it must not; on a cone the same hole cannot be placed at all; and now a hole is a short tube with mass in it, which is where the fingering chart comes from.

What the ladder owes now is the cutoff in this model. The lattice’s corner is the one quantity in this anchor that has always been computed from Benade’s formula for an infinite periodic lattice of identical holes — and the two halves of this essay say that a real instrument’s holes are neither identical nor periodic, by design and by a factor of two in diameter down one bore. The solver here carries every hole individually and could be swept over a real chart of stations and diameters, which would say whether a woodwind’s cutoff is one frequency at all or whether every fingering has its own. If it is the second, then the timbre changes that players describe across the break are a property of the fingering rather than of the instrument, and this collection has been quoting a single number for something that is a list.